Engineering FailureAnalysis, Vol 2, No. 4 pp. 307-320, 1995
Pergamon
Copyright © 1995 Elsevier Science Ltd
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DEVELOPMENTS IN THE ANALYSIS OF INTERACTING
CRACKS
R. JONES,* S. N. A T L U R I , t S. PITT* and J. F. WILLIAMS*
*Department of Mechanical Engineering, Monash University, Wellington Rd, Clayton, Victoria,
Australia, 3168; tCenter for Computational Modelling, Georgia Institute of Technology, Atlanta,
GA, U.S.A.; *Department of Mechanical and Manufacturing Engineering, Melbourne University,
Parkville, Victoria, Australia
(Received 19 June 1995)
Abstract--This paper presents an overview of the finite element alternating technique for the
analysis of interacting cracks. To illustrate the ease and accuracy of this method the technique
is used to analyse several problems associated with both widespread fatigue and multi-site
damage, a problem which is attracting worldwide attention. Whilst this paper presents an
overview of the technique for both two- and three-dimensional problems attention is focused
on three-dimensional problems. In particular, the interaction effects between two fully
embedded elliptical flaws and between two semi-elliptical surface flaws, and the effects of
crack proximity and crack aspect ratio on the stress intensity factors are presented. For
semi-elliptical surface flaws these results indicate that as the cracks approach each other the
position of the point on the crack front with the highest stress intensity factor shifts. This
subsequently suggests that surface cracks will tend to grow preferentially towards each other.
The same trend is evidenced for fully embedded cracks. However, in this case there is no shift
in the position of the maximum stress intensity factor. A discussion of the results in terms of
stress intensity magnificationfactors is also presented.
1. INTRODUCTION
The downturn in the global economy and the end of the "cold war" coupled with the
high acquisition costs associated with the purchase of modern military and civilian
aircraft has resulted in greater utilisation of existing aircraft fleets. This trend, in
operating existing aircraft approaching or beyond their intended design life, has been
reflected in an increasing number of structurally significant defects. The long service
life of "ageing" aircraft increases the possibility of a reduction, or loss, of structural
integrity d u e to fatigue. The importance of understanding and managing ageing
structures was highlighted by the failure of Aloha 737 on 28 April 1988. This failure
was essentially due to the linking, into one large crack, of numerous small cracks at a
number of fastener holes. This p h e n o m e n o n has subsequently been termed "rn*ulti-site
damage" (MSD). MSD presents unique problems-to both the aircraft maintenance
operator and the structural analyst. In essence, it occurs when groups of small cracks
appear at about the same time and are located in a c o m m o n area [1]. Even though
each crack, considered individually, may be safe the level of interaction can degrade
the damage tolerance of the structure beneath acceptable levels.
1.1. A n Australian perspective
1.1.1. F/A-18 laboratory tests. Although the p h e n o m e n o n was first observed in
civilian aircraft, recent Australian work [2] has found that MSD plays a major role in
determining the fatigue life of the F/A-18 aft bulk head, i.e. bulk head FS-488 [2]. In
this case the various fatigue test(s) undertaken by McDonnell Douglas in support of
the F/A-18 had indicated a large n u m b e r of potential hot spots, including the FS488
aft bulkhead flange, mold line and wing attachment lug. T o further assess the fatigue
performance of the FS488 aft bulkhead a full-scale fatigue test was performed in
Australia on a stand alone FS488 bulkhead. The test was performed t o primarily
307
308
R. JONES et al.
address the region of the wing attachment lug. In this test failure resulted from a
fatigue crack approximately 6 mm deep. However, post-failure inspection of the test
article revealed the presence of several hundred cracks within the critical region [2].
A fractographic evaluation of the specimen subsequently revealed that this population
of cracks exhibited similar crack growth rates and that the cracks remained very small
throughout the life of the component. Furthermore, the markings on the critical crack
could essentially be traced back to the beginning of the test (see [2] for more details).
The question thus arose as to the level of interaction of all of these cracks,
particularly their effect on the so-called "dominant flaw", and their combined effect
on both the "local" load path and the damage tolerance of the bulkhead.
As a result of this test it was concluded [2] that existing NDI techniques could not
be relied upon to find the critical crack. This conclusion highlighted the need to
develop both advanced NDI techniques as well as new analysis tools for the
assessment of structural integrity, particularly when the critical component contains
large numbers of interacting dimensional flaws.
1.1.2. Mirage IIIO cracking. The importance of understanding and managing
widespread fatigue damage has also gained international visibility. In Australia the
importance of understanding and managing widespread fatigue damage was highlighted by fatigue cracking in Mirage IIIO aircraft in service with the RAAF. In this
case full-scale fatigue testing of Mirage IIIO fighter aircraft wings at the Swiss Federal
Aircraft Factory (F + W) resulted in fatigue cracks at the innermost bolt holes along
the rear flanges of the main spars, with failure associated with cracking at bolt hole
number 9. The test was then continued with a starboard R A A F Mirage IIIO wing
(2190 h service) and a port Swiss Mirage IIIS wing (510 h service). After a relatively
short test life cracks were then found at the innermost bolt holes. Cracking was also
found in a number of other locations including (fuselage) frame 26. Further cracking
was experienced in the lower (tension) wing skin both at the fairing hole (nearest the
main spar) and at the fuel decant hole in the lower wing skin (see [2, 4] for more
details). In the R A A F wing a failure occurred from cracks which developed at the
single leg anchor nut (SLAN) rivet holes associated with bolt hole number.
Following this test crack indications were confirmed at identical locations, in the
main spar, in wings of the R A A F Mirage IIIO fleet. A detailed laboratory test
program was then undertaken and it subsequently found that the existence of the two
SLAN rivet holes meant that cracking, which was both complex and three-dimensional in nature, developed at the (two) SLAN holes as well as at the main bolt holes.
We were thus faced with the problem of distributed and interacting (three-dimensional)
flaws.
The F + W fatigue test undertaken in Switzerland also resulted in a major failure,
togeth~ with a number of nearby cracks, in the fuselage frame 26. The major crack
occurred at hole 18 on the left side of frame 26A and extended across the entire
flange and well into the web. Other cracks occurred in the region between holes 1
and 23 and included a 9 mm crack in the inner strap plate and small (less than 3 mm
long) cracks in holes 4, 7, 8, 18, 20 and 22. Prior to the major crack being detected
significant cracking had also occurred in the outer strap plate in the bottom of the
frame. At the time of the failure this region contained cracks with lengths of 40, 31,
20, 18 and 12 mm.
1.1.3. The R A A F Macchi recovery program. The importance of maintaining continued airworthiness was further highlighted by the November 1990 failure of an
R A A F Macchi aircraft (viz. A7-076) which suffered a port wing failure whilst in an
estimated 6 g manoeuvre. In this case it was found that failure was caused by fatigue
cracking originating from the "D17" rivet hole in the lower spar cap [5]. As a result
of this event a Macchi Recovery Program was initiated to determine the structural
condition of the fleet and to reassess the fatigue lives and management philosophies
of the main structural components. A tear down inspection program, which involved
309
Developments in the analysis of interacting cracks
10 post LOTEX wings, two fuselages, two fins and five horizontal tail planes being
destructively inspected (see [5] for details), was then undertaken.
Six of the wings showed significant cracking indications and of the, approximately,
1000 holes which were examined 100 revealed fatigue cracks, including major cracking
in the D series rivet holes. This program revealed the fatigue critical locations in the
centre section lower spar boom to be bolt holes 3-6 and 17-20. The flaws were
highly three-dimensional in nature and, from the failure investigation of aircraft
A7-076, the failure process had involved a number of interacting cracks. In this case
cracking had progressed from a web attachment fastener hole through the flange as
well as from the nearby wing attachment fastener (rivet) hole (Fig. 1). Fractographic
evidence also indicated multiple crack origins at the root of the rivet hole (Fig. 1). To
assist in establishing the critical crack size a detailed three-dimensional finite element
analysis of this cracking was performed. This established that, once the main crack
had grown past the flange, the stress intensity factor was essentially equal to its
fracture toughness. A more detailed description of this program, its underlying
philosophy and the Macchi Aircraft Structural Integrity Management Plan is given
in [5].
The mutual influence of the adjacent cracks increases the complexity of predicting
both the fatigue crack growth rate and the failure mechanisms. Unfortunately, few
solutions exist in the literature for interacting cracks in finite geometry bodies. For
interacting three-dimensional cracks the sparsity of solutions is even greater. The
challenge is therefore to develop analytical tools, which are simple to use, and which
will allow accurate and rapid assessment of structural integrity.
This paper discusses one such technique which is based on the finite element
alternating method, which has the advantage that the cracks need not be modelled
explicitly. Initially attention is focused on the technique as developed for three-dimensional problems [6, 7]. For three-dimensional problems the alternating finite element
technique makes extensive use of the analytical solution for a three-dimensional
elliptical flaw subject to arbitrary crack face loading. In this context one of the first
relevant solutions was obtained by Green and Sneddin [8], who solved the problem of
a penny-shaped crack, subject to uniform tension at infinity. Kassir and Sih [9] solved
the problem of uniform shear loading along the crack face and obtained an exact
solution in terms of two harmonic potential functions. The generalisation of these
Web attachment hole
Fati~
eracl
Other
t
Wing attachment hole
Fig. 1. Cross section view of failure surfaces in the Macchi spar.
M
310
R. JONES et al.
solutions for the cases when the crack surface was subjected to various degrees of
polynomial pressure distribution was the focus of many other important studies. The
work of Segedin [10] proposed the use of a certain type of ellipsoidal harmonics and
their partial derivatives which satisfy Laplace's equation. This approach was subsequently used by both Kassir and Sih [11] and Shah and Kobayashi [12].
In the work of Shah and Kobayashi [12], the contributions of the potentials to each
stress component on the crack surface were not linearly independent polynomial
functions, and so it was necessary to make a judicious choice for the potential
functions for each degree of loading. Their analysis considered only normal loading,
and, in addition, they were limited to just cubic polynomial distributions on the crack.
This work was subsequently generalised by Vijayakumar and Atluri [7] who considered arbitrary normal as well as shear loading. These authors derived expressions
for stress intensity factors near the flaw border, as well as for stresses in the far-field,
for these generalised loadings. The key to implementing this solution in the finite
element alternating technique was the development by Nishioka and Atluri [6] of a
general procedure for evaluating the necessary elliptic integrals.
The Schwartz Neumann alternating technique was originally applied to threedimensional fracture mechanics by Shah and Kobayashi [10]. However, this solution
suffered from a number of drawbacks, viz:
(i) The analytical solution was limited to just cubic loading on the crack face.
(ii) The solution was limited to media bounded by straight surfaces.
Whilst Smith and Kullgren [13] introduced the finite element method into the
alternating technique, again using only a cubic polynomial, Nishioka and Atluri [6]
were the first authors to use a full analytical solution for the complete polynomial.
This technique has subsequently been successfully applied to solve a range of
three-dimensional problems, viz thick plates [6], pressure vessels [14] and aircraft
attachment lugs [15], and was recently extended by Jones et al. [16], to include
arbitrary interacting cracks.
One major advantage of this technique is that by combining the finite element
method with the analytical solution we enable accurate results to be obtained using
only a relatively coarse mesh. Furthermore, since cracks are not modelled explicitly
this means that the crack configuration can be changed without complex remeshing,
and, as the crack geometry changes, it also removes the need for tedious remeshing.
Consequently, for problems associated with MSD the finite element alternating
method provides a very efficient and cost-effective method of analysis.
2. THE THREE-DIMENSIONAL FINITE ELEMENT ALTERNATING
METHOD
The basic steps in the finite element alternating technique can be explained as
follows (see [6, 7] for more details):
(1) The stresses in the uncracked body are first obtained using the finite element
method.
(2) The tractions, at the locations of the cracks, in the uncracked body are
determined.
(3) The tractions at the crack faces are then reversed and the problem is converted
into that of solving for the same cracked structure, where the crack faces are
subject to the (reversed) tractions determined in step (2), and where the outer
boundaries (surfaces) of the body are stress-flee.
(4) The analytical solution for a crack, in an infinite body, subjected to these
(reversed) tractions, is then used to calculate both the stress intensity factors and
the stresses at the boundary of the body.
Initially the resultant stresses (tractions), obtained from the infinite body solution, at
Developments in the analysis of interacting cracks
311
the external surfaces (boundaries) will not satisfy the requirement that the external
boundary be stress-free.
(5) If the resultant stresses, as calculated in step (4), at the external boundary are
below a user-defined tolerance then the required solution has been obtained.
(6) If not, the residual stresses (tractions) at the boundaries of the body are reversed,
and steps (1)-(5) are repeated, using these reversed stresses as a new load case.
(7) This iterative loop is continued until convergence occurs [see step (5)]. At this
stage the final solution is the sum of the individual solutions obtained in each of
the iterative loops.
2.1. Basic formulation: infinite geometry
As can be seen in the previous section the alternating finite element method makes
extensive use of the analytical solution for a three-dimensional crack, in an infinite
domain, subjected to arbitrary crack face loading. Consequently, for the sake of
completeness, we will now briefly outline this solution. Let us begin by defining u~
(a~ = 1, 2, 3) and (x~, (o~, fl = 1, 2, 3) as the displacements and stresses, respectively,
in a homogeneous, isotropic linear elastic solid. Hooke's law then tells us that
tx~= G(u~.~+
u~,~+ 1 -2V2vS°'~ur'r)"
(1)
In the absence of body forces the Navier displacement equations of equilibrium, in
rectangular Cartesian coordinates x, (a~ = 1, 2, 3), are
ut~,O~ + (1 -
2v)ue,,~t~=
0.
(2)
Let R denote the surface of the crack, which for simplicity we will define as lying in
the plane (z = ) x 3 = 0. As is often the case in fracture problems it is convenient to
consider the complementary problem in which the surface R of crack is subjected to
arbitrary tractions o3o,. Following the work of Trefftz the solution of this problem can
be expressed in terms of four harmonic functions ~p and q~ (a~ = 1, 2, 3) in the form
u~ = q~ + x3~P,~
(3)
so that the Navier displacement equilibrium equations are satisfied if
q~,~ + (3
-
4v)~,3
=
0.
(4)
The problem can be further simplified by expressing ~p and q~ in the form
=
=
(5)
and
t~l = (1 - 2v)(fl,3 + f3,1) - (3 - 4v)fl,3,
th = (1
-
2v)(f2,3
+
f3,2)
--
(3
--
= - ( 1 - 2v)(fm + f2a) - 2 ( 1
4v)f2,3,
-
(6)
v ) f 3 , 3.
Then the governing equations, viz:
~ , ~ = 0,
(7)
q,.,~ = o,
(8)
¢~,~ + (3 - 4v)~,3 = 0,
(9)
are satisfied if the three functions f~ ( a = 1, 2, 3) are harmonic.
For near elliptic flaws the cracks will he taken to be an ellipse in the ((z =) x3 = 0
plane. Let us now assume that the geometry of the flaw can be described by the
equation
x2 + xA = 1,
(10)
312
R. JONES
et al.
where al and a2 are the major and minor axes of the flaw respectively, i.e. a~ > a22.
For simplicity the geometry of the crack surface is more conveniently described in
ellipsoidal coordinates ~ (a~= 1, 2, 3), where these are the roots of the cubic
equation w(~) = 0, where
Xl
w(~) = 1
a21 + ~
a22 + ~
.
(11)
With this formulation it has been found that suitable forms for the potential functions
in the Trefftz's formulation are
(12)
f ~ --- E E C ~ , k , l F i d
k l
where
e k l -- a X~k+l
k aX 1
f:
[('O(s)]k+l+l
(13)
~/Q(s)'ds
In this notation re(s) = P(s)/Q(s) and
P(s) = (s - ~l)(s - ~2)(s - ~3),
(14)
Q(s) = s(s + a~)(s + a2).
(15)
If we denote the partial derivative of f~ with respect to xo (fl = 1, 2, 3) as f~,# we can
then write
(16)
f~,,# = E ~ C~,k,tFu,~,
k I
f~,~r = E E C~,k,,Fu,~r
(17)
k l
and
(18)
= EE
k
1
The components of the stresses can then be written in the form
(Yll----- 212[f3,11 + 2vf3,22 -- 2fl,31 -- 2vf2,32 + X3(V'?),ll],
022 = 2#[f3,22 +
2vf3,11
-
2f2,32 -
2vfl,31 + x3(V.f),22],
(19)
(20)
a12 = 212[(1 - 2v)f3,12 - (1 - v)(fL32 + f2,13) + x3(V.?),12],
(21)
033 = 212[--f3,33 + X3(V.?),33] ,
(22)
031---~ 212[--(1 -- V)fl,33 + V(fl,11 + f2,21) q- X3(V'?)13],
(23)
032 = 212[--(1 -- v)f2,33 "1" V(fl,12 + f2,22) "~" X3(V'?),23]"
(24)
Let us now express the tractions along the crack surface in the form
,.r(O)
3
=EE
i=0 j=O
A (i,j)
2m-2n+i~2n+j
m=O n=O
,
(25)
where a~ = 1, 2, 3 and the values of (i, j) specify the symmetries of the load with
respect to the axes of the ellipse (see [6, 7] for more details). The solution in terms of
the potential function is then assumed to be of the form
I
f~
----
1
M
k
E E E E rod)
"" ~ , k - l , l v
~t 2k-21+i,21+j"
i=O /=0
(26)
kffi0 l=O
Substituting for f~ and its various partial derivatives in Eqns (19)-(24) thereby
Developmentsin the analysisof interactingcracks
313
enables the coefficients of C to be obtained directly from the coefficients of A via a
simple matrix equation of the form (see [6] for details)
(A} = [ B ] ( c } .
NxN
Nxl
(27)
Nxl
Once the coefficients C have been determined, for the given crack surface loadings,
the stress intensity factors corresponding to these loads can be directly computed. For
the mode I problem the relevant equation takes the form
A1/4~, ~' ~ ~(--2)2k+i+Y(2k+i+j+l)!
K I = 8f¢
k=0 1=0
i=O j=O
x
1 {cosO]2k-21+i{sin0]2l+/_(i/)
/--/
a182\
al
/
\
a2
(28)
C'3,~-I,t,
|--I
/
where 0 is the elliptic angle and
A = a12sin2 0 + a2cos 2 0.
(29)
3. SEMI-ELLIPTICAL S U R F A C E FLAWS IN A SEMI-INFINITE B O D Y
3.1.
A single flaw
To illustrate this approach let us first consider a single semi-elliptical surface flaw in
an infinite body. For numerical purposes, in accordance with common practice, the
body was regarded as being infinite if its edges were further than 5 times the
semi-major crack length away from the crack centre. To model this situation with the
finite element alternating technique only one quadrant needs to be considered,
because of the two planes of symmetry inherent in this problem, and the resultant
(uncracked) mesh used consisted of a total of 216 20-noded brick elements, i.e. a
6 x 6 x 6 mesh. For the purposes of this analysis, the material properties were chosen
as E = 70 GPa and v = 0.33, and a remote tensile stress of 1000 MPa was assumed.
In the first set of cases considered the surface (half) crack length al = 18 mm, the
width 2b = 400 mm and the crack aspect ratio = 82/81, where a 2 is the (variable)
crack depth (Fig. 2). In this instance the maximum stress intensity factor occurs at
point Q on the crack face which is deepest into the solid. To illustrate the accuracy of
this technique the values obtained for the maximum stress intensity factor, for
different crack aspect ratios, were compared with two different sets of published
values [17, 18]. The result of this analysis can be seen in Fig. 3. The mean difference
between the finite element alternating technique values and those of Rooke and
z
/
~
/
Constantuniform f
m_y
Fig. 2. Surfaceflawgeometry.
R. JONES et al.
314
5.50E + 03
E
i
5.00E + 03
4.50E + 03
~ 4.00E + 03
que
3.50E + 03
.o 3.00E + 03
~.:~
2.50E + 03
I
2.0OE + 03
I
0.2
I
0.4
0.6
Crack aspect ratio (a2/at)
I
0.8
Fig. 3. Comparison of the maximum stress intensity factor vs crack aspect ratio (a2/al) for a
semi-elliptical surface flaw in a semi-infinite body with [17, 18]. In all cases al = 18 nun and
b = 200 ram.
Cartwright [17] was 3.8%, whilst that between the finite element alternating technique and those published by the FAA [18] was 4.6%. Given the uncertainties quoted
(approximately 5%) in the published values these differences were deemed to be well
within acceptable limits. Furthermore the values obtained also tend to lie between
those given in [17] and those given in [18].
3.2. Two identical surface flaws
Having established the validity of the model for one crack, the problem was
subsequently extended to investigate two interacting semi-elliptical surface cracks,
with their centres being a distance d apart, in a semi-infinite body (Fig. 4). In the
following cases the mesh density was essentially the same as described above, the
crack aspect ratio (a2/al) w a s fixed at 0.8 and the "crack separation distance" d was
held constant at 100 ram. The crack separation ratio (A = 2arid) was then allowed to
vary, and values of the mode I stress intensity factor were calculated at 5° increments
along the crack front.
The results of this analysis can be seen in Fig. 5. This graph indicates that, for two
interacting cracks, the point of maximum stress intensity factor was no longer the
point which lies deepest in the solid. The latter behaviour only occurred when the
cracks were relatively far apart (A < 0.3) and were (essentially) acting independently
of each other. As the crack spacing decreased, the position of the point with the
largest Kx gradually shifted away from the point furthest into the solid (0 = 90°)
2a - - I
I
A
A [
d
' _ _ [ B
2at
I
Fig. 4. Two identical interacting surface flaws.
Developments in the analysis of interacting cracks
1.0000E ÷ 0
,--
4
[
315
-
9.0000E + 0
•
~
~
.0000
k = 0.85
0
.9.o . ~
0 '~o~-x-x--x-x-x-x-x-x
- - . 0 - - k = 0.7
,~-A-x-x-x-x
6.0000E + 03 ~
- ~.o.e-~v-~-e-e-~-e-e-~-e
~e-'I-1~-~"
i
-
.oooo, ÷
4.~E
k =0.9
-
-
+ 03 -
-.-
-
~
3.00o0E+03'
_
_+.+.+_+_+_+_+_+_+-+-+-+-~+-+-+-.+-+
2.0000E + 03"
~ x ~
I
20
I
40
I
60
I
80
I
I~
= o.,
k=0.4
e
Angle (o) between the free surface and
position on the crack perimeter
k = 0.6
k = 0.2
- - + - - - k --0.1
Fig. 5. Variation in maximum stress intensity factor with elliptic angle, 0 < 0 < 90, for one of
a pair of semi-elliptical surface flaws in a semi-finite body. In each case a2/al = 0.8.
towards the other crack (0 = 0°). Physically this means that, as crack spacing
decreases, interaction effects become more pronounced and the cracks begin to grow
preferentially towards each other rather than progressing further into the bulk of the
solid.
A more common way of expressing these results is to use a normalised stress
intensity factor, called the stress intensity magnification factor, Fro, defined by
K (O)
Vm =
/
Oo
.x/"a2(a
E(k) v al
,
(30)
sin 2 0 + a cos 2 0) 1/4
where E(k) is the complete elliptic integral of the second kind, k 2 = (a 2 - a 2 ) /
a 2, and 0 is the elliptic angle. The expression in the denominator is in fact the stress
distribution for a single crack in an infinite elastic solid subject to uniform tension at
infinity. The stress magnification factor is useful as it provides an indication of the
percentage increase in the stress intensity factor due to the interaction effect of the
other crack.
The magnification factors for the previous s e t of c a s e s
(a2/a I = 0 . 8 and
d = 100mm) were calculated by first employing the finite element alternating
technique to evaluate the stress intensity factors on a single semi-elliptical surface
flaw. The results of this analysis can be seen in Fig. 6. This graph clearly illustrates
that, in all cases, the magnification factor was greatest at the free surface of the solid.
As expected, the magnification factors increase as the cracks get closer together.
However, the rate of increase was more pronounced at the free surface than at the
point on the crack front deepest into the solid.
To investigate the effect of crack aspect ratio on the stress intensity factor, and on
the magnification factors, attention was then focused on the point of the crack which
was deepest into the solid. In all cases the distance between the crack centres was
fixed at 100 ram. Two different crack separation ratios were considered: viz 3. = 0.5
and 3. = 0.8. As d was fixed at 100 mm the first case implies that al = 25 mm whilst
the second implies a I = 40 mm. The values of a 2 were then allowed to vary. The
relationship between stress intensity factor (at 0 = 90 °) and crack aspect ratio is
illustrated in Fig. 7. It must be remembered that the position of the maximum stress
intensity factor was different for these two curves. Figure 7 shows that as the cracks
get larger 0.e. as a2 increases) the stress intensity factor at 0 = 90° also increases.
The corresponding magnification factors were calculated in the manner described
R. J O N E S et al.
316
1
.
2
2
~
1.20
1.18
--
~=0.9
,
_-08
t,4
.
--o--- k = 0 . 5
1.0 e - ~ - o - o - e - - * - l - - . - t , II
0
I
-- .
I
.
.
.
.
I
60
.
It
I
80
20
40
Angle (*) between the free surface and
position on the crack perimeter
I
100
k = 0.2
--*--- kffiO.l
Fig. 6. Variation in magnification factors versus elliptic angle, O, for a pair of semi-elliptical
surface flaws in a semi-infinite body. In each case a2/al = 0.8.
8.00E + 03
if" 7.00E + 03
6.00E + 03
,- 5.00E + 03
2
4.00E + 03
"~ 3.00E + 03
o"
.-= 2.00E+03
,,
•
~, = 0.8
1.00E + 03
0.00E + O0
0
I
0.1
I
0.2
I
0.3
0.4
0.5
Crack aspect ratio
0.6
0.7
I
0.8
Fig. 7. Variation in m a x i m u m stress intensity factor with crack aspect ratio for one of a pair of
identical semi-elliptical surface flaws in a semi-infinite body. In all cases d = 100 ram.
above, and the results can be seen in Tables 1 and 2. For both values of the crack
separation ratio (4 = 0.5, 0.8) there was little change in the magnification factor at
0 = 90°. In the situation of two cracks it would therefore appear that, at the point
where the cracks protrude deepest into the solid, the crack separation is a more
crucial factor than the crack shape.
Table 1. Magnification factors for a pair of semielliptical surface flaws in a semi-infinite solid
(0 = 90 °, X = 0.8).
a2/al
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
k
E(k)
0.995
0.98
0.95
0.92
0.87
0.8
0.714
0.6
1.01599
1.02859
1.06047
1.08793
1.12845
1.17848
1.235 68
1.29842
KI
(N m m -3R)
3.49E
5.02E
6.13E
6.89E
7.41E
7.73E
7.89E
7.93E
+
+
+
+
+
+
+
+
03
03
03
03
03
03
03
03
Magnification
factor
1.0025
1.0299
1.0585
1.0586
1.0548
1.0491
1.039 5
1.0269
Developments in the analysis of interacting cracks
317
Table 2. Magnification factors for a pair of semielliptical surface flaws in a semi-infinite solid
(0 = 90°; ~ = 0.5)
a2/al
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
k
E(k)
0.99
0.96
0.91
0.84
0.75
0.64
0.51
0.36
1.01599
1.02859
1.06047
1.08793
1.12845
1.17848
1.23568
1.29842
KI
(N mm -3/~)
2.75E
3.52E
4.59E
5.15E
5.55E
5.82E
6.00E
6.11E
+ 03
+ 03
+ 03
+ 03
+ 03
+ 03
+ 03
+03
Magnification
factor
0.9984
0.99035
1.00277
0.9996
0.9994
0.9998
1.0000
1.0001
4. THE FINITE ELEMENT ALTERNATING METHOD IN TWO DIMENSIONS
In the previous section we saw that the three-dimensional finite element alternating
method provided an efficient and cost-effective analysis method. To illustrate the
simplicity and accuracy of this approach for two-dimensional problems, which differs
only in that a different analytical solution is used, let us consider a flat plate,
containing multiple cracks, under uniform uniaxial tension. For the purposes of this
investigation the plate was assumed to be 10mm thick and have dimensions
100 x 100mm. The plate was also assumed to be an aluminium alloy with
E = 71 GPa and v = 0.32, and the remote (uniform) stress was taken to be 68.9 MPa.
For the uncracked problem a relatively coarse mesh consisting of nine elements was
used (see Fig. 8) with the (various) cracking states under consideration located in
element 5. For these test cases two crack configurations were considered, viz: (a) a
row of four horizontal cracks, and (b) a column of four cracks, and the resultant
stress intensity factors were compared with standard handbook values [17].
4.1.
Case (a)
In this test case a row of four equal length cracks were "placed" along the x-axis.
The length of each crack was designated as 2a and the spacing between the crack
centres as 2b (Fig. 9). The results reveal that the maximum stress intensity factors
occur at locations A and B as shown in Fig. 9. The resultant stress intensity factors,
normalised by dividing by Ko = o~(rra), are shown in Fig. 10 together with the
y-Axis
Uniformly distributed uniaxial load
7
8
5
1
6
~x-Axis
2
r
Fig. 8. Schematic diagram showing the details of the mesh used.
318
R. J O N E S et al.
2b
2b
I
I
I
II
2a
2b
I
AB
I
II
II
2a
2a
I
2a
Fig. 9. Nomenclature used in the first series of test cases. The cracks are identical and are
equally spaced.
1.6-
1.4 ~1.2 "o
!--
0
o
o
rl
0.60.40.20
0
I
0.1
I
0.2
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
I
0.8
I
0.9
a/b
"Published
values
oCalculated values
Fig. 10. Stress intensity factors for a horizontal row of four cracks.
values obtained from [17]. In general the values of KI/K o agree with the published
values to within 2.6%.
4.2. Case (b)
The second series of test cases involved a stack of four horizontal cracks. The
geometry for this and the nomenclature adopted are as shown in Fig. 11. In reference
[17] two sets of stress intensity factors are published for this crack configuration. The
first of these two sets are for the tips of the outermost cracks. These cracks have the
highest stress intensity factors. The other series of results refer to the tips of the inner
cracks. These cracks are partly shielded by the outer ones and hence the stress
intensity factors at these locations are lower than for the outermost crack.
As in the previous test case the stress intensity factors, thus calculated, were
normalised with respect to Ko and the resultant values, plotted as a function of a/c,
are shown in Figs 12 and 13. From Figs 12 and 13 it can be seen that the results agree
with the published values to within 0.5%.
5. CONCLUDING REMARKS
This paper has outlined the alternating finite element technique as applied to both
three- and two-dimensional fracture problems. For interacting cracks the mutual
D e v e l o p m e n t s in the analysis of interacting cracks
319
y-Axis
Uniformly distributed uniaxial load
0
.......
i
.......
i
m
.~ x-Axis
9
- - ~ 2a
Fig. 11. Crack configuration for the second series of test cases. " o " designates an outer crack
and " r ' an inner crack.
E
~
E
E
[]
[]
0.8
13
13
0.6
'°f
0.4
0.2
0
I
0.2
I
0.4
I
I
0.6
0.8
a/c
o Calculated values
I
1.0
I
I
1.2
I
! .4
1.6
APublished values
Fig. 12. Stress intensity factors for the outermost crack in a vertical stack of four horizontal
cracks.
0.8
1.01
~9 0.6
E E
i
j
E
0.4
0.2
0
I
0.2
I
0.4
I
0.6
I
0.8
a/¢
o Calculated values
I
1.0
I
1.2
I
1.4
I
1.6
•Published values
Fig. 13. Stress intensity factors for the middle crack in a stack of four horizontal cracks.
influence of the adjacent cracks significantly increases the complexity of predicting
both the fatigue crack growth rates and the failure mechanisms. The challenge was to
develop analytical tools, which are simple to use, and which will allow accurate and
rapid assessment of structural integrity. This paper discusses one such technique based
of the finite element alternating method [6, 7, 19]. One advantage of this method is
that the cracks need not be modelled explicitly. To illustrate its simplicity particular
attention has been focused on the effects of interacting surface and embedded cracks
in a three-dimensional solid. In this case it has been shown that the stress intensity
factors depend upon a number of variables including:
(i)
crack configuration, viz whether the cracks are surface flaws or are fully
embedded,
(ii) individual crack geometry, viz the shape of the crack itself, as measured by the
crack aspect ratio (a2/al),
320
R. JONES et al.
(iii) the separation between the cracks, as determined by Z (= 2aJd). In most
instances the crack interaction effects can be neglected until the cracks are
closely spaced.
From the problems detailed in this paper the following trends are evident:
(a) For a single semi-elliptical surface flaw in a semi-infinite body the point with the
highest stress intensity factor is always the point on the crack face lying deepest
in the solid. However, for two interacting surface flaws this situation changes and
the cracks interact in such a way as to promote crack growth towards each other,
i.e. link-up.
(b) For two interacting semi-elliptical surface flaws the crack separation ratio has a
greater influence on the stress intensity factor at the point deepest in the solid
than does the crack aspect ratio.
(c) For a single fully embedded elliptical flaw the point with the highest stress
intensity factor lies on the semi-minor axis. This is still the case for two
interacting fully embedded elliptical flaws and also promotes crack growth
towards each other.
REFERENCES
1. W. R. Hendrieks, The Aloha Airlines Accident--a New Era for Aging Aircraft, Structural Integrity of
Aging Airplanes, pp. 152-165, Springer-Verlag, Berlin (1991).
2. S. Barter and G. Clark, Proceedings of International Aerospace Congress 1991, Melbourne, 12-16 May
1991, Vol. 1, pp. 35-43 (1991).
3. J. Y. Mann, Defence Science and Technology Organisation, Aeronautical Research Laboratory,
Structures Technical Memo 361 (1983).
4. R. Jones, Bonded Repair of Aircraft Structure (edited by A. A. Baker and R. Jones), Chap. 4,
Martinus Nijhoff, The Hague (1988).
5. G. D. Young, Macchi MB326H Aircraft Structural Integrity Management Plan, Issue 1, RAAF
Logistics Systems Agency, Logistics Command, Melbourne (1994).
6. T. Nishioka and S. N. Atluri, Engng Fracture Mech. 17,247-268 (1983).
7. K. Vijayakumar and S. N. Atluri, J. Appl. Mech. 48 (1981).
8. A. E. Green and I. N. Sneddin, Proc. Camb. Phil. Soc. 46, 159-163 (1950).
9. M. K. Kassir and G. C. Sih, J. Appl. Mech. 33,601-611 (1966).
10. C. M. Segedin, Report 67-3, Department of Aeronautics and Astronautics, College of Engineering,
University of Washington (1967).
11. M. K. Kassir and G. C. Sih, Mechanics of Fracture (edited by G. C. Sih), Vol. 2, Nordhoff
International (1975).
12. R. C. Shah and A. S. Kobayashi, Fracture Mech. 3, 71-96 (1971).
13. F. W. Smith and T. E. Kullgren, AFFDL-TR-76-104, Air Force Flight Dynamics Laboratory (1977).
14. T. Nishioka and S. N. Atluri, J. Pressure Vessel Technol. 104,299-307 (1982).
15. T. Nishioka and S. N. Atluri, AIAA J. 21,749-757 (1983).
16. S. Hammond, R. Jones, J. F. Williams and S. N. Atluri, Proceedings of the 6th Australian Aeronautical
Conference, 20-23 March 1995, pp. 173-177, Institution of Engineers Australia, National Conference
Proceedings NCP 95/1 (1995).
17. D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity Factors, Her Majesty's Stationary
Office, London (1976).
18. T. Swift, Federal Aviation Administration Course Notes. also see USAF Damage Tolerant Design
Handbook: Guidelines for the Analysis and Design of Damage Tolerant Aircraft Structures, US
Department of Commerce, National Technical Information Service, AD-A153 161, Part 1 (1984).
19. S. N. Atluri and P. Tong, Structural Integrity of Aging Airplanes (edited by S. N. Atluri, S. G.
Sampath and P. Tong), pp. 15-35, Springer-Verlag, Berlin (1991).